PHYSICAL REVIEW B VOLUME 36, NUMBER 8 15 SEPTEMBER 1987-I

Scaling relations between the two- and three-dimensional polaronsfor static and dynamical properties

F. M. Peeters and J. T. Devreese*

(Received 19 March 1987)

Scaling relations are presented which allow one to obtain static and dynamical physical quanti-

ties of the two-dimensional (2D) polaron out of known results for the 3D polaron. These scaling

relations are exact within second-order perturbation theory and are approximately valid (on a l%%uo

level) for higher orders in the electron-phonon coupling constant. A formal generalization ofthese scaling relations to the n-dimensional polaron is given.

From the 1950's on the polaron problem has been anattractive field of research. In recent years there hasbeen a renewed interest, both from theoreticians and ex-perimentalists, in the concept of polarons. Besides its in-trinsic interest in the area of solid-state physics, the po-laron problem has become a reference problem for test-ing various methods of approximation because itrepresents a comparatively simple, physically realisticexample of the interaction of a nonrelativistic particlewith a quantized field. During the years the polaronproblem has become a "classic" problem in quantumfield theory.

Polarons are quasiparticles consisting of an electron(or a hole) which is surrounded by its polarization cloud.The electron is moving in a polar semiconductor or anionic crystal. The physical properties of polarons inthree-dimensional systems have been studied' experimen-tally and theoretically ever since Landau introduced theconcept. Theoretically a variety of techniques have beenused from which the Feynman path-integral methodturns out to be the most successful one.

With the advent of new fabrication techniques, like,e.g. , molecular-beam epitaxy, metal-organic chemical-vapor deposition etc. , it has become possible to growstructures in which the electrons are localized in spacein one or two directions. This leads to two (2D) or evenone-dimensional (1D) electron systems and, consequent-ly, if these systems are made out of polar semiconduc-tors, one has 2D or 1D polarons. Over the last fewyears 2D polarons have been studied by several experi-mental and theoretical groups.

In actual systems the polaron will not be exactly twodimensional; it is more justified to speak about quasi-two-dimensional polarons. The thickness of the 2D elec-tron layer perpendicular to the layer is nonzero.

In the present paper we are interested in ideal 2D sys-tems. One-polaron properties will be studied, and wewill neglect the many-electron nature of the system. Thecalculated polaron quantities are expected to result in anupper bound for the practical polaron eAect which willbe measured in real systems. Screening and nonzero lay-er width vary from one system to another and are notuniversal. Therefore the following assumptions will be

I'

made: we consider a strict 2D model, neglect interfacephonons and screening, and assume that the Frohlichcontinuum polaron model is valid. This will allow us toprovide a unifying and comprehensive picture of theproperties of 2D polarons where the energy scale is setby the bulk-LO-phonon frequency.

A series of scaling relations will be developed whichwill allow us to obtain several important physical quanti-ties of the 2D polaron from known 3D-polaron results.The 3D polaron has been studied over the years, and anextensive polaron literature' is available for them. Thesescaling relations will be generalized to the n-dimensional(nD) polaron.

The electron-phonon interaction will be described bythe Frohlich Hamiltonian

+ g&~~a~akP2mb

k

where p and r are the conjugate coordinates of the elec-tron with band mass mz and al, (ak) are the creation(annihilation) operators for a LO phonon with wave vec-tor k and energy Ace~. A parabolic energy band for theelectron is assumed, the system is considered isotropicand the LO phonons are assumed to be dispersionless,i.e., co~ ——coLO. The electron-phonon coupling is charac-terized by

(2)

V ( 3) is the volume (surface area) of the system, and ais called the Frohlich electron-phono n coupling con-stant. Units are used such that R =mb ——cu Lo

——1. TheHarniltonian for the 2D system was first derived by Sak.

For completeness we mention that in a recent paper

36 4442 1987 The American Physical Society

36 BRIEF REPORTS

we already found that the polaron ground-state energysatisfies the approximate scaling relation

3wE2D(a) =—'-Ego a

3 4

where E2D is the ground-state energy of the 2D polaron.This scaling relation is exactly valid to second order inthe electron-phonon coupling constant ~ It is also exactlysatisfied by the Feynman polaron result. As a conse-quence the earlier obtained results' ' for the 2D-polaron ground-state energy are trivially obtained fromthe known 3D-polaron result. The scaling relation is ap-proximately valid for higher orders in a. Note also thatthe absence' of a discontinuous self-trapping transitionfor the 3D optical polaron implies that similarly no suchtransition is expected for the 2D optical polaron (seealso Ref. 15). The generalization of the scaling relation(3) to nonzero temperature is straightforward. One thenfinds that the electron-phonon contribution to theHelmholtz free energy satisfies the same scaling relationas the polaron ground-state energy.

An important characteristic of the Frohlich polaron isits (linear) response to an oscillating electric field. Thisresponse provides us information on the excited states ofthe polaron. The response is given by the inverse of theimpedance function and the real part of this quantitycorresponds to the mobility for small frequencies (i.e.,IM ((coLo ) and to the optical absorption for frequenciesin the infrared region (i.e. , co) co„o). Applying the Feyn-man path-integral technique Feynman, Hellwarth, Id-dings, and Platzman (FHIP) derived the impedancefunction Z(co). The inverse of this function is given by

are the parameters of the Feynman polaron model whichare determined by a variational calculation of the pola-ron free energy. The weak-coupling results are obtainedif we take the limit v ~w.

Inserting Eq. (2) into Eq. (4) we find that the memoryfunctions for the 2D and 3D polarons are connected inthe following way:

This leads to a similar scaling relation for the impedancefunction

3nZzD ( a; cd ) =Z iD a; co

As an example we plot in Fig. 1 the optical absorption atzero temperature for ca=3 in the case of a 3D and a 2Dpolaron as obtained from the FHIP theory. It is ap-parent that for one polaron in 2D the polaron effects aremore pronounced. In 2D the relaxed excited states al-ready show up for a=3, while in 3D no such states arepresent for such a relative low electron-phonon couplingconstant. The polaron in 2D is more strongly bound toits polarization cloud than in 3D for the same value of

The polaron mass at zero temperature can be obtainedfrom the impedance function' ' in the following way:

ReX(co )= 1 —lcm

1

Z (co)

1

co —X(co )(4)

0.5

where X(co) is called the memory function and corre-sponds to a force-force correlation function' whose timeevolution is governed by a Liouville operator which isprojected onto the Liouville space which is orthogonal tothe velocity operator x. The result is (z=co+ie withe~+0)

0.4—

K=3T=O

20

X(z)= —I dt( l —e"') ImS(t), 0.3—

with the function

g(t)= /2'~

yg~

k

X t [I+n(~k)]e ' +n(ccik )e ' I, (6a)

and

wD(t)=2v 2

2

Pfi

2 2

[I e"'+4n(U) sin ( —'Ut )], —2v 2

FREQUENCY ( ~IIELD)

16 20

where P is the inverse temperature and n (co) the occupa-tion number of the LO phonons. The parameters v, w

FIG. 1. Optical absorption for a 2D and a 3D polaron atzero temperature and for an electron-phonon coupling constant(x=3.

BRIEF REPORTS 36

where mb is the band mass. From the scaling relationfor the memory function we also have a scaling relationfor the polaron mass

m,*D(a)

(mb )pD

3~m 3D tX

(mb )3D(10)

3'P~D(a ) =P3D

4

where the band mass in a 2D system, (mi, )zo, may bedifferent from its 3D counterpart, (mb)3D. The validityrange of this scaling relation is the same as for Eq. (3).In Fig. 2 we have plotted the polaron mass as functionof the coupling constant (a) for different approximatecalculations of m* which all satisfy the scaling relation(10). The scale for the 2D-polaron result is given on thetop of the figure while the coupling constant for the 3Dpolaron is shown below on the figure.

It has been argued and demonstrated' that the zero-frequency linear mobility as obtained from Eq. (4) doesnot lead to the correct result. The relaxation-time ap-proximation (RTA) has to be used (which corresponds tothe correct order for the limits: lim .Olim„o) in orderto obtain the correct result. In Ref. 19 this has beengeneralized (within the Feynman polaron model) to gen-eral values of e. Following Ref. 19 one can easily provethe following scaling relation for the mobility

Recently the present authors have extended theFrohlich Hamiltonian to arbitrary space dimensions.The electron-phonon coupling is then characterized by

2n —3/2 (1/ j(n —1)

E„D(a)= —E3D( A„a ), (13a)

m„*o(a) m;D( A„a)(mb )3D (mb )3D

Z„o(a ) =Z3D ( 3„a),

(13b)

(13c)

and

p.D(a ) =Iu3D( &.a )

where we defined

I((n —1)/2) 3&rrI (n/2) 2n

(13d)

which has been obtained from the condition that theFrohlich Hamiltonian in n —1 dimensions can be ob-tained from the n-dimensional (nD) result by integratingout the n th dimension. V is the volume of the n-dimensional crystal and I (x) is the I' function. In asimilar way as above we are able to derive scaling rela-tion for the nD polaron

which is valid within RTA and the FHIP theory.

R(20)02 0.4 0.6 0.8 1 '1.2 1.4 1.6

{ I

0 01 0.2 0.3 0.4 /

E

E

Ul

E

C)

(QC)CL

2

u (30I

FIG. 2. Polaron mass as function of u for different approxi-mations (RSPT stands for second-order Rayleigh-Schrodingerperturbation theory and LLP for the Lee-Low-Pines canonicaltransformation approach'). The scale for the 2D polaron isgiven at the top of the figure while the scale for the 3D polaronis given at the bottom.

All the quantities listed here and for which approxi-mate scaling relations have been found, i.e., the ground-state energy (and also the free energy), the polaron mass,the impedance function, and the mobility, have the prop-erty that they do not have any preferential direction inspace. If no such spherical symmetry is present no scal-ing relations are found. The consequences of this obser-vation are twofold. First, only those approximationswhich preserve this spherical symmetry will satisfy thescaling relation. This is true for second-order perturba-tion theory, for approximations based on the Lee-Low-Pines variational calculation, for approximations basedon the Feynman polaron model, for approximationswith a general quadratic action, etc. Secondly, onlyphysical quantities with no preferential direction inspace can have the above type of scaling relation. Thisis the reason why there is no scaling relation for theenergy-momentum relation, for the properties of a pola-ron in a magnetic field (i.e. , ground-state energy andcyclotron resonance), nonlinear transport (i.e.,velocity —electron-field characteristics), etc.

That the energy-momentum relation [E(p)] does notsatisfy a scaling relation has been demonstrated recently.In Ref. 20 it was found that the 2D E(p) relation isqualitatively diferent from the corresponding 3D rela-tion. Around the LO-phonon continuum, i.e. , atE(p) =E(p =0)+ficoLo, the 3D E(p) relation is continu-ous (although its first derivative is discontinuous ') butin 2D an energy gap with zero width (in the k ~ ao lim-it) is found. Also in the E(p) relation a preferentialdirection is present, i.e., the direction in which the elec-tron moves.

36 BRIEF REPORTS

An idea about the validity range of the scaling rela-tions for arbitrary o. values will be given now. For thatpurpose take as an example the ground-state energy toorder & which in 3D is exactly given byE30(a) = —a —0.015 919m . After application of thescaling relation we find E2'D'" (a) = —(n/2)a—0.058917o. , which should be compared to the exactresult EzD(cz) = —(rr/2)a —0.063 97a . The coefficientto second order in perturbation theory is exactly givenby this scaling relation while the a term is underes-timated by 8%. Even for a=0.2 Eg'"g(a=0. 1) givesthe energy correct within 0.06%. In the large couplinglimit the dominant term to the 3D ground-state energyis E3D ———0. 108 513m, which after application of thescaling relation gives Ezo"" ———0.4016o. and whichcompares to the exact result E2D ———0.4047' . There-fore the scaling relation underestimates the result with0.7'7o. We took the ground-state energy as an examplebecause a lot of accurate results are know, but from ourprevious studies we expect that a similar reasoning holdsfor the polaron mass, the mobility, etc. From the abovediscussion we may infer that the scaling relation for gen-

eral values of u hold on a 1% level.In conclusion, we have found that (1) the ground-state

energy (or the free energy when T&0), (2) the effectivemass, (3) the linear mobility, and (4) the optical absorp-tion for a polaron moving in two dimensions are con-nected to the corresponding results in three dimensionsby simple scaling relations. These scaling relations areexact (i) to second order in the electron-phonon couplingconstant and (ii) for harmonic type of approximationslike, e.g. , approximations based on the Feynman polaronmodel. The scaling relations are approximately valid forhigher orders in the electron-phonon coupling constant.These scaling relations are formally generalized to then-dimensional optical polaron problem.

This work is partially supported by FKFO (Fondsvoor Kollektief Fundamenteel Onderzoek, Belgium),Project No. 2.0072.80. One of us (F.M.P.) acknowledgesfinancial support from the Belgium National ScienceFoundation.

Also at University of Antwerp (Rijksuniversitair CentrumAntwerpen), B-2020 Antwerpen and University of Technolo-gy, NL-5600 MB Eindhoven, The Netherlands.

'See, for example, Polarons and Excitons, edited by C. Kuperand G. Whitfield (Oliver 4 Boyd, Edinburgh, 1963); Pola-rons in Ionic Crystals and Polar Semiconductors, edited by J.T. Devreese (North-Holland, Amsterdam, 1972); Polaronsand Excitons in Polar Semiconductors and Ionic Crystals,edited by J. T. Devreese and F. M. Peeters (Plenum, NewYork, 1984).

R. P. Feynman, Phys. Rev. 97, 660 (1955).3R. P. Feynman, R. W. Hellwarth, C. K. Iddings, and P. M.

Platzman, Phys. Rev. 127, 1004 (1962). For an actual calcu-lation of the optical absorption we refer to J. T. Devreese, J.De Sitter, and M. Goovaerts, Phys. Rev. B 5, 2367 (1972).

4K. K. Thornber and R. P. Feynman, Phys. Rev. B 1, 4099(1970).

5K. K. Thornber, Phys. Rev. B 3, 1929 (1971).F. M. Peeters, Wu Xiaoguang, and J. T. Devreese, Phys. Rev.

B 33, 3926 (1986).7H. Frohlich, H. Pelzer, and S. Zienau, Philos. Mag. 41, 221

(1950).8J. Sak, Phys. Rev. 8 6, 3981 (1972).Wu Xiaoguang, F. M. Peeters, and J. T. Devreese, Phys. Rev.

B 31, 3420 (1985).W. J. Huybrechts, Solid State Commun. 28, 95 (1978); M.

Matsuura, ibid. 44, 1471 (1982).''G. A. Farias, N. Studart, and O. Hipolito, Solid State Com-

mun. 43, 95 (1982); E. L. Bodas and O. Hipolito, Phys. Rev.B 27, 6110 (1983).

' N. Tokuda, J. Phys. C 15, 1353 (1982).S. Das Sarma and B. A. Mason, Ann. Phys. (N.Y.) 163, 78(1985).

' F. M. Peeters and J. T. Devreese, Phys. Status. Solidi B 112,219 (1982).

'~B. Gerlach and H. Lowen, Phys. Rev. B 35, 4291 (1987).' F. M. Peeters and J. T. Devreese, Phys. Rev. B 28, 6051

(1983).' J. T. Devreese, L. F. Lemmens, and J. Van Royen, Phys.

Rev. B 2, 1212 (1977).F. M. Peeters and J. T. Devreese, Phys. Status Solidi B 115,539 (1983).

' F. M. Peeters and J. T. Devreese, Solid State Phys. 38, 81(1984).F. M. Peeters, P. Warmenbol, and J. T. Devreese, Europhys.Lett ~ 3, 1219 (1987).

'D. M. Larsen, Phys. Rev. 144, 697 (1966).G. Hohler and A. Mullensiefen, Z. Phys. 157, 159 (1959).J. Roseler, Phys. Status Solidi 25, 311 (1968).

24D. M. Larsen, Phys. Rev. 172, 967 (1968).25S. J. Miyake, J. Phys. Soc. Jpn. 38, 181 (1975).